Embedded eigenvalues of Sturm Liouville operators

In this work we study the behavior of embedded eigenvalues of Sturm-Liouville problems in the half axis under local perturbations. When the derivative of the spectral function is strictly positive, we prove that the embedded eigenvalues either disappear or remain fixed. In this case we show that local perturbations cannot add eigenvalues in the continuous spectrum. If the condition on the spectral function is removed then a local perturbation can add infinitely many eigenvalues.     

Construction of exponential Liouville field operators for closed string models

Operators for arbitrary exponentials exp(λφ) of a periodic Liouville field φ(τ,σ) are represented iteratively by an infinite power series in terms of a periodic scalar free field. Necessary quantum corrections of the Liouville operators with respect to their classical expressions are fixed by conformal covariance and locality. Canonical commutation relations for the Liouville field quantities are valid when the canonical quantization of the scalar free field is imposed. A quantum correction of the energy momentum tensor can be avoided thus preserving the conformal invariance of the Liouville theory.     

甲醛和过氧化氢分子振动光谱的路径积分刘维尔动力学模拟

本文采用路径积分刘维尔动力学计算偶极矩时间导数的关联函数来得到红外光谱,通过与真实振动频率相比,发现路径积分刘维尔动力学可以比较忠实地描述振动动力学过程中的核量子效应,能够准确刻画系统温度变化和同位素取代效应引起的振动光谱变化.     

Generalizations of the Virasoro algebra and matrix Sturm–Liouville operators

We study the series of Lie algebras generalizing the Virasoro algebra introduced in [V. Yu, Ovsienko, C. Roger, Functional Anal. Appl. 30 (4) (1996)]. We show that the coadjoint representation of each of these Lie algebras has a natural geometrical interpretation by matrix differential operators generalizing the Sturm–Liouville operators.     

On the Relationship of the Spectra of the Self-adjoint Operator and its Liouville Operator

In this paper, we study the self-adjoint extensions of the Liouville operator and correct the relationship of absolutely and singular continuous spectra between a Hamiltonian and its corresponding Liouvillian. We also give the relationship of the essential and discrete spectra between the two operators.     

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

Spectra of massive and massless QCD dirac operators: A novel link

We show that integrable structure of chiral random matrix models incorporating global symmetries of QCD Dirac operators (labeled by the Dyson index beta = 1,2, and 4) leads to emergence of a connection relation between the spectral statistics of massive and massless Dirac operators. This novel link established for beta-fold degenerate massive fermions is used to explicitly derive (and prove the random matrix universality of) statistics of low-lying spectra of QCD Dirac operators in the presence of SU(2) massive fermions in the fundamental representation ( beta = 1) and SU(N(c)>/=2) massive adjoint fermions ( beta = 4). Comparison with available lattice data for SU(2) dynamical staggered fermions reveals a good agreement.     

Spectra of tensor operators adapted to nonstandard basis. Qualitative features of clustering

Spectra of irreducible tensor operators for the chain of group G ? O(3) are investigated from the qualitative point of view. Conditions are found for clustering of the eigenvalues of tensor operators in the high-J limit. Approximate formulas are proposed which permit the calculation of the relative positions of the clusters. Centrifugal distortion of spherical tops, crystal-field splitting, and low-frequency bending vibrations of nonrigid molecules composed of an atom and rigid symmetrical core are considered as physical examples. Clustering of the F^(kJJ′) coefficients of Moret-Bailly are qualitatively explained as well.     

Coulomb integrals in Liouville theory and Liouville gravity

The four-point correlation function has been studied in Liouville field theory. If one of the fields is degenerate, such a function is described in terms of Coulomb integrals. Some nontrivial relations for these integrals have been found that can be used to obtain new exact results in conformal field theory. In particular, a four-point correlation function has been calculated in minimal quantum gravity. The result agrees with the results obtained recently by different methods [A. A. Belavin and A. B. Zamolodchikov, JETP Lett. 82, 7 (2005); Theor. Math. Phys. 147, 729 (2006); A. B. Zamolodchikov, Theor. Math. Phys. 142, 183 (2005); I. K. Kostov and V. B. Petkova, Theor. Math. Phys. 146, 108 (2006)]. The text was submitted by the authors in English.     

Quantum Liouville Equation

The symmetrized product of quantum observables is defined. It is seen as consisting of ordinary multiplication followed by application of the superoperator that orders the operators of coordinate and momentum. This superoperator is defined in the way that allows obstruction free quantization of algebra of observables as well as introduction of operator version of the Poisson bracket. It is shown that this bracket has all properties of the Lie bracket and that it can substitute the commutator in the von Neumann equation leading to quantum Liouville equation.     

Super Liouville black holes

We describe in superspace a classical theory of of two-dimensional (1,1) dilaton supergravity coupled to a super-Liouville field, and find exact super black hole solutions to the field equations that have non-constant curvature. We consider the possibility that a gravitini condensate forms and look at the implications for the resultant spacetime structure. We find that all such condensate solutions have a condensate and/or naked curvature singularity.     

Fractional generalization of Liouville equations

In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered as a normalization condition for systems in fractional phase space. The interpretation of the fractional space is discussed.     

Nonperturbative model of Liouville gravity

We formulate nonperturbative 2D gravity in the framework of Liouville theory. In particular, we express the specific heat of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. We recognize the relevant divisors on moduli spaces and write the integrands in terms of the Liouville action. We evaluate the integrals (rational intersections) and show that satisfies the Painlevé I.